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THEORY OF INDICES 1.a m xa n =a m+n 2.(a m ) n =a mn 3.a m /a n =a m-n 4.(ab) m =a m b m 5.a 0 =1 6.a x/y =y th root of a x = y √a x 7.a 1/p =p th root of a 8.(ab/c) m =a m b m /c m 9.a ∞ =∞ 10.a -∞ =0 Find the least number with which you multiply 882, so that the product may be a perfect square. First find the factors of 882. 882 = 2 × 3 × 3 × 7 × 7 Now, 882 has factors as shown above, ‘3’ repeated twice, ‘7’ repeated twice and ‘2’ only once. So when one more factor ‘2’ is used, then it becomes a perfect square. 882 × 2 = (2 × 2) × (3 × 3) × (7 × 7) The least number required is ‘2’

Characteristics of square roots of numbers 1.If a square number ends in ‘9’, its square root is a number ending in’3’ or ‘7’. 2. If a square number ends in ‘1’, its square root is a number ending in’1’ or ‘9’. 3. If a square number ends in ‘5’, its square root is a number ending in’5’ 4. If a square number ends in ‘4’, its square root is a number ending in’2’ or ‘8’. 5. If a square number ends in ‘6’, its square root is a number ending in’4’ or ‘6’. 6. If a square number ends in ‘0’, its square root is a number ending in ‘0’ Ex: 1)√529=23    √729=27    √1089=33   √ 1369=37, etc 2)√121= 11    √361= 19    √961 =31      √81= 9,etc 3)√625=25 √1225=35 √2025=45 & so on 4)√484=22     √64=8    √1024=32    √784=28 & so on 5)√196=14     √256=16    √576=24    √676=26 & so on 6)√10...

Characteristics of square numbers 1. A square cannot end with an odd number of zeros 2. A square cannot end with an odd number 2, 3, 7 or 8 3. The square of an odd number is odd 4. The square of an even number is even. 5. Every square number is a multiple of 3 or exceeds a multiple of 3 by unity. Ex. 4 × 4 = 16 = 5 × 3 + 1 5 × 5 = 25 = 8 × 3 + 1 7 × 7 = 49 = 16 × 3 + 1 6. Every square number is a multiple of 4 or exceeds a multiple of 4 by unity. Ex. 5 × 5 = 25 = 6 × 4 + 1 7 × 7 = 49 = 12 × 4 + 1 7. If a square numbers ends in ‘9’, the preceding digit is even. Ex. 7 × 7 = 49 ‘4’ is the preceding even numbers 27 × 27 = 729 ‘2’ is the preceding even numbers. Characteristics of square roots of numbers 1. If a square number ends in ‘9’, its square root is a number ending in’3’ or ‘7’. 2. If a square number ends in ‘1’, its square root is a number ending in’1’ or ‘9’. 3. If a square number ends in ‘5’, its square root is a number ending in’5’ ...

Problems: 1. If a number when divided by 296 gives a remainder 75, find the remainder when 37 divides the same number. Method: Let the number be ‘x’, say ∴x = 296k + 75, where ‘k’ is quotient when ‘x’ is divided by ‘296’ = 37 × 8k + 37 × 2 + 1 = 37(8k + 2) + 1 Hence, the remainder is ‘1’ when the number ‘x’ is divided by 37. 2. If (2^32)+1 is divisible by 641, find another number which is also divisible by ‘641’. Method:                                           NOTE:^-POWER OF NUMBER Consider 2^96+1 = (2^32)^3 + 1^3 = (2^32 +1)(2^64-2^32 +1) From the above equation, we find that 296+1 is also exactly divisible by 641, since it is already given that 232+1 is exactly divisible by ‘641’. 3.3. If m and n are two whole numbers and if m^n = 25. Find n^m, given that n ≠ 1 m^n = 25 = 5^2 ∴m = 5, n = 2 ∴n^m = 2^5 = 32 4.A number when successively divided by 9, 11 a...

Number System Test of divisibility: 1. A number is divisible by ‘2’ if it ends in zero or in a digit which is a multiple of ‘2’i.e. 2,4, 6, 8. 2. A number is divisible by ‘3’, if the sum of the digits is divisible by ‘3’. 3. A number is divisible by ‘4’ if the number formed by the last two digits, i.e. tens and units are divisible by 4. 4. A number is divisible by ‘5’ if it ends in zero or 5 5. A number is divisible by ‘6’ if it divisible by ‘2’ as well as by ‘3’. 6. A number is divisible by ‘8’ if the number formed by the last three digits, i.e, hundreds tens and units is divisible by ‘8’. 7. A number is divisible by ‘9’ if the sum of its digit is divisible by ‘9’ 8. A number is divisible by ‘10’ if it ends in zero. 9. A number is divisible by ‘11’ if the difference between the sums of the digits in the even and odd places is zero or a multiple of ‘11’. LCM: LCM of a given set of numbers is the least number which is exactly divisi...

1. H.C.F. and L.C.M. of Decimals Step 1 Make the same number of decimal places in all the given numbers by suffixing zero(s) if necessary. Step 2 Find the H.C.F./L.C.M. of these numbers without decimal. Step 3 Put the decimal point (in the H.C.F./L.C.M. of step 2) leaving as many digits on its right as there are in each of the numbers 2. L.C.M. and H.C.F. of Fractions L.C.M = L.C.M. of the numbers in numerators/H.C.F. of the numbers in denominators H.C.F. = H.C.F. of the numbers in numerators/L.C.M. of the numbers in denominators 3. Product of two numbers = L.C.M. of the numbers  x H.C.F. of the numbers 4. To find the greatest number that will exactly divide x, y and z. Required number = H.C.F. of x, y and z. 5. To find the greatest number that will divide x, y and z leaving remainders a, b and c, respectively. Required number = H.C.F. of (x – a), (y – b) and (z – c). 6. To find the least number which is exactly divisible by x, y and z. Required number = L.C....

Common Factor A common factor of two or more numbers is a number which divides each of them exactly. For example, 4 is a common factor of 8 and 12. Highest common factor Highest common factor of two or more numbers is the greatest number that divides each one of them exactly. For example, 6 is the highest common factor of 12, 18 and 24. Highest Common Factor is also called Greatest Common Divisor or Greatest Common Measure. Symbolically, these can be written as H.C.F. or G.C.D. or G.C.M., respectively Methods of Finding H.C.F. I. Method of Prime Factors Step 1 Express each one of the given numbers as the product of prime factors. [A number is said to be a prime number if it is exactly divisible by 1 and itself but not by any other number, e.g. 2, 3, 5, 7, etc. are prime numbers] Step 2 Choose Common Factors. Step 3 Find the product of lowest powers of the common factors. This is the required H.C.F. of given numbers. Illustration 1 Find the H.C.F. of 70 and 90. Solu...